# 1: Fundamentals of Signal Processing

## 1.0 INTRODUCTION

The Fourier transform is fundamental to seismic data analysis. It applies to almost all stages of processing. A seismic trace represents a seismic wavefield recorded at a receiver location. The digital form of a seismic trace is a time series which can be completely described as a discrete sum of a number of sinusoids – each with a unique peak amplitude, frequency, and a phase-lag (relative alignment). The analysis of a seismic trace into its sinusoidal components is achieved by the forward Fourier transform. Conversely, the synthesis of a seismic trace from the individual sinusoidal components is achieved by the inverse Fourier transform. A brief mathematical review of the Fourier transform is given in Appendix A.

Seismic data processing algorithms often can be described or implemented more simply in the frequency domain than in the time domain. In Section 1.1, the one-dimensional (1-D) Fourier transform is introduced and some basic properties of time series in both time and frequency domains are described. Many of the processing techniques – single- and multichannel, involve an operand (seismic trace) and an operator (filter). A simple application of Fourier analysis is in the design of zero-phase frequency filters, typically in the form of band-pass filtering.

The two-dimensional (2-D) Fourier transform (Section 1.2) is a way to decompose a seismic wavefield, such as a common-shot gather, into its plane-wave components, each with a certain frequency propagating at a certain angle from the vertical. Therefore, the 2-D Fourier transform can describe processes like migration and frequency-wavenumber (f-k) filtering. A common application of f-k filtering is the rejection of coherent linear noise by dip filtering, and attenuation of multiples based on velocity discrimination between primaries and multiples in the f-k domain (Section 6.2).

*figures&tables*

*contents*

*georef*

*references*

*related*

### Figures & Tables

### Contents

# Seismic Data Analysis: Processing, Inversion, and Interpretation of Seismic Data

The Classical Greeks had a love for wisdom –

It came down to us as philo sophia.

And I have a passion for the seismic method –

Let this be an ode to philo seismos.

O how sweet it is –

Listening to the echos from the earth.

The seismic method has three principal applications:

(a) Delineation of near–surface geology for engineering studies, and coal and mineral exploration within a depth of up to 1 km: The seismic method applied to the near-surface studies is known as engineering seismology.

(b) Hydrocarbon exploration and development within a depth of up to 10 km: The seismic method applied to the exploration and development of oil and gas fields is known as exploration seismology.

(c) Investigation of the earth’s crustal structure within a depth of up to 100 km: The seismic method applied to the crustal and earthquake studies is known as earthquake seismology.

This book is devoted to application of the reflection seismic method to the exploration and development of oil and gas fields.

Conventional processing of reflection seismic data yields an earth image represented by a seismic section which usually is displayed in time. Figure I-1 shows a seismic section from the Gulf of Mexico, nearly 40 km in length. Approximate depth scale indicates a sedimentary section of interbedded sands and shales down to 8 km. Note from this earth image a salt sill embedded in the sedimentary sequence. This allocthonous salt sill has a rugose top and a relatively smooth base. Note the folding and faulting of the sedimentary section above the salt.

The reflection seismic method has been used to delineate near-surface geology for the purpose of coal and mineral exploration and engineering studies, especially in recent years with increasing acceptance. Figure I-2a shows a seismic section along a 500-m traverse across a bedrock valley with steep flanks. The lithologic column based on borehole data indicates a sedimentary sequence of clay, sand, and gravel deposited within the valley. The bedrock is approximately 15 m below the surface at the fringes of the valley and 65 m below the surface at the bottom of the valley. The strong reflection at the sediment-bedrock boundary is a result of the contrast between the low-velocity sediments above and the high-velocity Precambrian quartz pegmatite below.

The reflection seismic method also has been used to delineate the crustal structure down to the Moho discontinuity and below. Figure I–2b shows a seismic section recorded on land along a 15-km traverse. Based on regional control, it is known that the section consists of sediments down to about 4 km. The reflection event at 6.5–7 s, which corresponds to a depth range of 15–20 km, can be postulated as the crystalline basement. The group of reflections between 8–10 s, which corresponds to a depth range of 25–35 km, represents a transition zone in the lower crust – most likely, the Moho discontinuity, itself.

Common-midpoint (CMP) recording is the most widely used seismic data acquisition technique. By providing redundancy, measured as the fold of coverage in the seismic experiment, it improves signal quality. Figure I–3 shows seismic data collected along the same traverse in 1965 with single-fold coverage and in 1995 with twelve-fold coverage. These two different vintages of data have been subjected to different treatments in processing; nevertheless, the fold of coverage has caused the most difference in the signal level of the final sections.

Seismic data processing strategies and results are strongly affected by field acquisition parameters. Additionally, surface conditions have a significant impact on the quality of data collected in the field. Part of the seismic section shown in Figure I-4 between midpoints A and B is over an area covered with karstic limestone. Note the continuous reflections between 2 and 3 s outside the limestone-covered zone. These reflections abruptly disappear under the problem zone in the middle. The lack of events is not the result of a subsurface void of reflectors. Rather, it is caused by a low signal-to-noise (S/N) ratio resulting from energy scattering and absorption in the highly porous surface limestone.

Surface conditions also have an influence on how much energy from a given source type can penetrate into the subsurface. Figure I-5 shows a seismic section along a traverse over a karstic topography with a highly weathered near-surface. In data acquisition, surface charges have been used to the right of midpoint A, and charges have been placed in holes to the left of midpoint A. In the absence of source coupling using surface charges, there is very little energy that can penetrate into the subsurface through the weathered near-surface layer. As a result, note the lack of coherent reflections to the right of midpoint A. On the other hand, improved source coupling using downhole charges has resulted in better penetration of the energy into the subsurface in the remainder of the section.

Besides surface conditions, environmental and demographic restrictions can have a significant impact on field data quality. The part of the seismic section shown in Figure I-6 between midpoints A and B is through a village. In the village, the vibroseis source was not operated with full power. Hence, not enough energy penetrated into the earth. Although surface conditions were similar along the entire line, the risk of property damage resulted in poor signal quality in the middle portion of the line.

Other factors, such as weather conditions, care taken during recording, and the condition of the recording equipment, also influence data quality. Almost always, seismic data are collected often in less-than-ideal conditions. Hence, we can only hope to attenuate the noise and enhance the signal in processing to the extent allowed by the quality of the data acquisition.

In addition to field acquisition parameters, seismic data processing results also depend on the techniques used in processing. A conventional processing sequence almost always includes the three principal processes – deconvolution, CMP stacking, and migration.